A parallel repetition theorem for entangled projection games
We study the behavior of the entangled value of two-player one-round projection games under parallel repetition. We show that for any projection game of entangled value , the value of the -fold repetition of goes to zero as , for some universal constant . If furthermore the constraint graph of is expanding we obtain the optimal . Previously exponential decay of the entangled value under parallel repetition was only known for the case of XOR and unique games. To prove the theorem we extend an analytical framework introduced by Dinur and Steurer for the study of the classical value of projection games under parallel repetition. Our proof, as theirs, relies on the introduction of a simple relaxation of the entangled value that is perfectly multiplicative. The main technical component of the proof consists in showing that the relaxed value remains tightly connected to the entangled value, thereby establishing the parallel repetition theorem. More generally, we obtain results on the behavior of the entangled value under products of arbitrary (not necessarily identical) projection games.
Relating our relaxed value to the entangled value is done by giving an algorithm for converting a relaxed variant of quantum strategies that we call “vector quantum strategy” to a quantum strategy. The algorithm is considerably simpler in case the bipartite distribution of questions in the game has good expansion properties. When this is not the case, the algorithm relies on a quantum analogue of Holenstein’s correlated sampling lemma which may be of independent interest. Our “quantum correlated sampling lemma” generalizes results of van Dam and Hayden on universal embezzlement to the following approximate scenario: two non-communicating parties, given classical descriptions of bipartite states , respectively such that , are able to locally generate a joint entangled state using an initial entangled state that is independent of their inputs.
Two-player one-round games arise naturally in many areas of theoretical computer science. They are prominent in complexity theory, where they are a powerful tool for the study of constraint satisfaction problems, and in cryptography, where they give a polyvalent abstraction used to establish the security of many two-party primitives. They have also recently proven a very convenient framework for the study of some of the deepest issues in quantum mechanics, giving a novel viewpoint on the decades-old study of Bell inequalities [BCP14], which are linear inequalities that must be satisfied by any family of distributions that can be generated locally according to the laws of classical mechanics, but can be violated if the distributions are allowed to be generated using quantum entanglement.
A game is specified by finite sets of questions, , of answers, a probability distribution on pairs of questions , and an acceptance criterion which states, for every possible pair of questions , which pairs of answers are valid. The most basic quantity associated to a game is its value. This can be defined operationally as the maximum success probability of two cooperating, but spatially isolated, players in the following game: a trusted party (the “referee”) selects a pair of questions according to , and sends to the first player (“Alice”) and to the second (“Bob”). Each player replies with an answer , and the players win the game if and only if .
Remarkably, the precise definition of the value depends on the physical theory used to model the a priori vague assumption that the players be “spatially isolated”. Under classical theory, isolated players are fully described by the (possibly randomized) functions they each apply to their respective question in order to determine their answer, and this interpretation leads to the classical value val of the game. In contrast, in quantum theory isolated players are allowed any set of strategies that can be implemented by performing local measurements on a shared entangled state. The resulting value is called the entangled value and denoted . Clearly for every game it holds that , and it is the discovery of Einstein, Podolsky and Rosen [EPR35] (formalized by Bell [Bel64], simplified by Clauser et al. [CHSH69] and experimentally verified by Aspect et al. [AGR81]) that there exist games for which the inequality is strict; indeed there are families of games for which but [Raz98, Ara02]. One can go even further and consider the non-signaling value , which corresponds to players allowed to reproduce any bipartite correlations that do not imply signaling. Here again , and there are games, such as the CHSH game [CHSH69], for which the inequality is strict.
One of the most fundamental questions one may ask about two-player games is that of the behavior of the value under product. Given games and , their product is defined as follows: the question and answer sets are the cartesian product of those from and ; the distribution on questions is the product of the distributions, and the acceptance criterion the and of those of and of . How does the value of relate to that of and ? While it is clear that each of the three values defined above satisfies , the reverse inequality, although intuitive, does not hold in general. In particular, simple constructions of games are known such that [FL92]; similar constructions exist for [CSUU08] and [KR10].
In spite of these examples, one may still ask for the behavior of , for “large” values of . This is known as the parallel repetition question: given a game such that , does there exist a such that whenever and ? If so, what form does take? Can it be approximately linear in the vicinity of ? Answering this question is of importance for many of the applications of two-player games. In cryptography, parallel repetition is a basic primitive using which one may attempt to amplify the security guarantees of a given protocol; in the study of Bell inequalities it can be used e.g. to amplify gaps between the quantum and non-signaling values; in complexity theory it is an important tool for hardness amplification.
For the case of the classical value, a sequence of works [Ver94, Fei91, FK00] over the course of a decade led to the breakthrough by Raz [Raz98], who was the first to provide a positive answer for general games: Raz showed that one can always take , where , are universal constants. Subsequent work focused on obtaining the best possible value for (the best known for general games is [Hol09]) and on removing the dependence on the size of the answer alphabet for specific classes of games [Rao08, BRR09, RR12]. For the case of the no-signaling value, Holenstein showed one can always take for some constant [Hol09].
In contrast, for the case of the entangled value in spite of its importance the question is very poorly understood. Strong results are known for some very special classes of games such as XOR games [CSUU08], for which repetition is exact (one can take ) and unique games [KR10] (for , where is a universal constant). However, both these results, as well as related results motivated by cryptographic applications [HR09], rely on the formulation of the entangled value as a semidefinite program, a characterization that is not believed to extend to more general games. Additional results are known but they only apply to specific games often originating from cryptography [MPA11, TFKW13]. Prior to this work the most general results known came from [KV11], where it is shown that a specific type of repetition inspired by work of Feige and Kilian [FK00], in which the original game is mixed with “consistency” and “free” games, reduces the entangled value at a polynomial rate: provided , the value of “Feige-Kilian” repetitions of behaves as for some small . (See “related work” below for additional discussion of more recent results that appeared after the initial completion of this work.)
A recent work of Dinur and Steurer [DS13] introduces a new approach to the parallel repetition question, focused on the case of projection games. A projection game is one in which the referee’s acceptance criterion has a special form: for any pair of questions , any answer from the second player determines at most one valid answer for the first player. Projection games are among the most interesting and widely-studied type of games. In particular, any local constraint satisfaction problem can be made into a projection game as follows: one player is asked for an assignment to all variables appearing in a constraint chosen at random, and the other is asked for an assignment to one of its variables. This simple transformation easily generalizes to convert any two-player game into a projection game , while essentially preserving the value: (see Claim 5). In particular, if one is only interested in “amplifying the gap” between and one can first map to and then consider the parallel repetition of itself, and this justifies the predominant role played by projection games in classical complexity theory. This transformation, however, may decrease the entangled value arbitrarily whenever the optimal strategy for the players requires the use of entanglement (though we show that it can never increase the value by too much; see Claim 5 for precise bounds). Nevertheless, many of the games studied in quantum information, such as the CHSH game [CHSH69] or the Magic Square game [Ara02] are projection games.
The approach of [DS13] is based on the introduction of a relaxation of the game value, denoted . This relaxation can be defined for any game (we give the definition in Section 1.2 below), and it is perfectly multiplicative. Moreover, for the case of projection games turns out to remain closely related to val, thus leading to a parallel repetition theorem. Although such a theorem already follows from Raz’s general result [Raz98], this arguably simpler approach matches the best parameters currently known [Rao08], which are known to be optimal [Raz08]. In addition, it yields new results for repetitions of games with small value and the case of few repetitions, which has implications for the approximability of the label cover and set cover problems.
1.1 Our results
We extend the analytical framework introduced in [DS13] to the case of the entangled value . As a consequence we obtain the following main theorem on the parallel repetition of the entangled value of projection games.
There exists constants such that the following holds. For any projection game ,
Although we do not attempt to fully optimize the constant , the value that come out of our proof is . For the case of expanding games (see definition in Section 2.2) we obtain the optimal .
Parallel repetition results for the classical value were originally motivated by the study of multi-prover interactive proofs [FRS88], and our result is likewise applicable to the study of classes of multi-prover interactive proofs with entangled provers. Letting denote the class of languages having -prover -round interactive proofs in which completeness holds with unentangled provers, but soundness holds even against provers allowed to share entanglement, Theorem 1 implies that for any . This is because any protocol in can be put into a form where the verifier’s test is a projection constraint by following the reduction already discussed above, and described in Claim 5; this will preserve both perfect completeness (for classical strategies) and soundness bounded away from (for quantum strategies). Prior to our work it was not known how to amplify soundness to exponentially small without increasing the number of rounds of interaction. It follows from [IV12, Vid13] that , but very little is known about the -prover class .
We believe that our results should find applications to a much wider range of problems. Going beyond the application to the parallel repetition question, our main contribution is the development of a precise framework in which general questions about the behavior of the value under product can be studied. This framework constitutes a comprehensive extension of the one introduced in [DS13] for the study of the classical value: as in [DS13], we introduce a relaxation of the entangled value, prove that it is perfectly multiplicative, and show that it remains closely related to . We find it remarkable that the framework from [DS13], introduced in a purely classical context, would find such a direct extension to the case of the entangled value. We hope that the tools developed in this extension will find further applications to the proof of product theorems in areas ranging from cryptography to communication complexity. Even though at a technical level the setting can appear quite different, some of the ideas put forth here could also prove useful to further removed areas such as the additivity conjecture for the minimum output entropy of quantum channels [AHW00, HW08, Has09].
We turn to a more detailed explanation of our framework, hoping to highlight precisely those tools and ideas that may find further application.
1.2 Proof sketch
In order to explain our approach it is useful to first review the framework introduced in [DS13] for the study of the classical value.
The starting point in [DS13] consists in viewing games as operators acting on the space of strategies. In this language a strategy is simply a vector of non-negative reals indexed by pairs of possible questions and answers: is the probability that the strategy provides answer to question . To any game one can associate a matrix such that, formally, the success probability of strategies for the players equals the vector-matrix-vector product . The value of the game is then the norm of when viewed as an operator on the appropriately normed spaces of strategies.
The first crucial step taken in [DS13] consists in relaxing the value of a game to the value of a symmetrized version of the game, which we call the square of the game (this notation will be made precise in Section 2.2); we will denote the latter value by . In the square of a game , the referee first samples a question for the first player as in . He then independently samples two questions and for the second player according to the conditional distribution. The players in are sent and respectively. They have to provide answers and such that there exists an such that both is a valid answer to in , and is a valid answer to . Note that now is in general not a projection game, even if was. In particular, treats both players symmetrically, and it turns out that we may always assume that they both apply the same strategy. For the special case of projection games it is not hard to show that the value of the game and that of its square are quadratically related:
Indeed, using the algebraic language introduced above, the first inequality follows from the Cauchy-Schwarz inequality and the second is an easy observation.
The second step consists in observing that the application of the operator corresponding to the product , where and are arbitrary projection games, can be decomposed as a product . Starting with a strategy for , the result of applying to is a new vector which no longer satisfies the strict normalization requirements of strategies. Understanding the new normalization leads to a further relaxation of , denoted , in which the optimization is performed over the appropriate notion of “vector strategies”, which intuitively are vectors that can be obtained by applying game operators to strategies. With the correct definition, it is easy to show that
The third and last step, which constitutes most of the technical work in [DS13], consists in showing that is a good approximation to . This is done using a rounding procedure, by which a vector strategy associated with a large is mapped back to an actual strategy for the square game that also has a high value, thus serving as a witness for the value being large as well. Altogether we get a bound on the value of as a product of a bound on the value of and a bound on the value of . Repeated application of (2) then leads to the following chain of inequalities (where the last approximate equality hides a polynomial dependence)
proving the parallel repetition theorem.
Our goal now is to extend the above sketch to the case of the entangled value . There is good reason for optimism. In contrast to most classical proofs that appear in the study of classical two-player games (such as those that go into Dinur’s proof of the PCP theorem [Din07], or earlier approaches to parallel repetition [Ver94, FK00, Raz98]), which are often information-theoretic or combinatorial in nature, the analytic (one could say linear-algebraic) framework introduced in [DS13] seems much better suited a priori to an extension to the quantum domain. Indeed, quantum strategies themselves are objects that live in -dimensional complex vector space: instead of a vector of non-negative reals (giving the probability of answering to question , for every possible and ), a strategy is now a vector of -dimensional positive semidefinite matrices that describe the measurement to be performed upon receiving any question . The normalization condition is for every , a constraint dictated by the formalism of measurements in quantum mechanics. Note that taking we recover classical strategies; quantum mechanics allows to be arbitrarily large.
At an abstract level, going from the classical to the entangled value thus solely requires us to think of the game as an operator acting on a bigger space of strategies, “enlarging” the non-negative reals to the space of -dimensional positive semidefinite matrices. This operation is easily realized by “tensoring with identity”, .
It remains to show how to extend each of the steps outlined above. The first step consists in obtaining an analogue of (1). As in the classical case the second inequality is easy, and follows by observing that, if is a quantum strategy in then is a valid strategy for the first player in (this notation will be made precise in Section 2.2.) The first inequality in (1) is slightly more subtle. Although it can be shown directly by applying a suitable matrix version of the Cauchy-Schwarz inequality, we note that it can also be proven using known properties of a widely used construction in quantum information theory, the pretty-good measurement (PGM) [HW94, HJS96]. As it turns out, the relaxation precisely corresponds to replacing the first player’s optimal choice of strategy in by a near-optimal choice obtained from the pretty-good-measurement derived from the post-measurement states, on the first player’s space, that arise from the second player’s measurements. As a consequence, (1) extends verbatim:
Next we need to find an appropriate notion of vector strategy and corresponding relaxed value . Here we are helped by the “operational” interpretation of a vector strategy as the result of the application of a game operator to a strategy meant for the product of several games. With the suitable generalization of the definition of classical vector strategies (see Definition 10) we also obtain an analogue of (2) for :
Even though this is not directly needed for our purposes, we note that itself is perfectly multiplicative (see Lemma 8 for the easy proof).
Finally, and most arduous, is to relate the relaxation back to the value of the square game, . In the classical case this involves rounding vector to actual strategies. In the quantum case rounding has to be performed synchronously by the players, and will necessarily involve the use of an entangled state. Intuitively, upon receiving their respective questions in the players need to initialize themselves in an entangled state that corresponds to the post-measurement state that they would be in, conditioned on having given a particular pair of answers to a given pair of questions in the game from which the vector strategy is derived (recall that, informally, vector strategies are the result of applying a game operator to a strategy meant for the product of two or more distinct games).
In case the bipartite distribution of questions in the game has good expansion properties we can show that this conditioned state is roughly the same regardless of the respective questions received by each player in , so there is a way for players to renormalize their measurements and proceed. For the non-expanding case the states can differ significantly from question to question. Nevertheless, we can show that based on their respective questions the players are able to agree on classical descriptions of two close states that they respectively wish to be in.
Since the questions are not known to the players a priori, they need to generate the appropriate entangled states “on the spot”, from an initial shared entangled state that is independent from and . Our new “quantum correlated sampling” lemma allows the players to do just this: given classical descriptions of respectively, they are able to generate a joint entangled state from an initial shared universal “embezzlement state” [vH03] independent of or , without any communication. The lemma can be seen as a quantum variant of Holenstein’s correlated sampling lemma [Hol09], as well as a “robust” extension of the results of van Dam and Hayden on universal embezzlement states [vH03]. We discuss this lemma and related works in more detail in Section 5.
All steps having been extended, we obtain a direct generalization of the chain of inequalities (3) to the case of entangled strategies:111We note however that the approximate equality that we obtain in the quantum case, although it suffices for our application to parallel repetition, is weaker than the one from [DS13]. In particular, it is probably not tight.
1.3 Additional related work
Although few general results are known, the question of the behavior of the entangled value of a two-player game or protocol under parallel repetition arises frequently. It plays an important role in recent results on device-independent quantum key distribution [HR09, MPA11] and related cryptographic primitives [TFKW13]. The latter work considers parallel repetition of a game with quantum messages, a setting which is also the focus of [CJPP11]. The approach of [CJPP11] builds upon [JPPG10], who relate the (classical) value of a two-player one-round game to the norm of the game when viewed as a tensor on the space . This is similar to our starting point of viewing games as operators acting on strategies, except that it considers the game as a bilinear form rather than an operator; the two points of view are equivalent. This perspective enables the authors to leverage known results on the study of tensor norms in Banach space (resp. operator space) theory to derive results on the classical (resp. entangled) value. To the best of our knowledge this connection has not led to an alternative approach to proving parallel repetition for general classes of games, although partial results were obtained in [CJPP11] for the special case of the entangled value of rank-one quantum games.
After the completion of this work two new results established an exponential parallel repetition theorem for two-player one-round games with entangled players in which the distribution on questions is a product distribution. In [CS14] it is shown that the entangled value of games in which the distribution on questions is uniform decreases as
Very recently Jain et al. [JPY13] extended the result to arbitrary product distributions on the questions, while also removing the dependence on the number of questions: they obtained the bound
Both results are based on the use of information-theoretic techniques. They are incomparable to ours, as they apply to games in which the acceptance predicate is general but the input distribution is required to be product. In addition, both bounds above have a dependence on the number of answers in the game; while for the case of the classical value such a dependence is necessary [FV02], for the entangled value it is not yet known whether it can be avoided.
1.4 Open questions
We briefly mention several interesting open questions. There still does not exist any parallel repetition result that applies to the entangled value of general, non-projection two-player one-round games, and it would be interesting to investigate whether our techniques could lead to (even relatively weak) results in the general setting. The case of three players is also of interest, and no non-trivial parallel repetition results are known either in the classical or quantum setting. In fact, the closely related question of XOR repetition of three-player games is known to fail dramatically even for the classical value [BBLV12].
Organization of the paper.
We start with some important preliminaries in Section 2. There we introduce the representation of games and strategies that is used throughout the remainder of the paper. In Section 3 we introduce the two relaxations of the entangled value sketched in the introduction and give a more detailed overview of our proof. In Section 4 we prove the main technical component of our work, the relation between and . Finally, in Section 5 we state and prove the quantum correlated sampling lemma.
We thank Attila Pereszlényi for comments on an earlier version of this manuscript.
We identify , the set of linear operators from to , with the set of matrices with complex entries: if then its matrix has entries , where , range over the canonical bases for , respectively, and we use the bra-ket notation to denote column vectors and row vectors , where denotes the conjugate-transpose. We also write for . The space is a Hilbert space for the inner product . We let be the operator norm of , its largest singular value. A state is a vector with norm .
The following simple calculation, sometimes known as Ando’s identity, will be useful.
Let , be two operators and a bipartite state with Schmidt decomposition , where the are non-negative reals. Then
where and the transpose is taken in the bases specified by the and . In particular, if for every , is positive semidefinite and (4) evaluates to .
The proof follows by direct calculation, expanding the left-hand side of (4) using the Schmidt decomposition of and the right-hand side using the definition of . ∎
We state a matrix analogue of the Cauchy-Schwarz inequality; we include a proof for completeness (see also [Pis03, p.123]).
For any and operators , ,
Let be unit vectors with Schmidt decomposition and . For any and ,
where , , and . Applying the Cauchy-Schwarz inequality once more,
Since (5) holds for any and , the claim is proved. ∎
2.2 Games and strategies
A two-player game is specified by finite question sets and , finite answer sets and , a distribution on , and an acceptance criterion . We also write for . The game may also be thought of as a bipartite constraint graph, with vertex sets and , edge weights , and constraints on each edge . We will write for the marginal distribution of on , and its marginal on . (We omit the subscripts and when they are clear from context.) We also often write to mean that is distributed according to the conditional distribution . The size of is defined as .
In this paper we focus on projection games, which are games for which the acceptance criterion is such that for every there is at most one such that . Equivalently, for every edge the associated constraint is a projection constraint such that is the unique such that if it exists, and a special “fail” symbol otherwise. When the edge is clear from context we will write to mean that . We also write to mean that there exists an such that and .
Given a projection game , let be the weighted adjacency matrix associated with the square of : is the matrix whose -th entry equals . Let be the diagonal matrix with the degrees on the diagonal, and the normalized Laplacian associated with the square of . We say that a family of games , where has size , is expanding if the second smallest eigenvalue of is at least a positive constant independent of .
Projection games as operators.
Let be a two-player projection game. We will think of as a linear operator defined as follows:
In other words, for , let denote the value of at the coordinates indicated by basis vectors and . Then
Note that here we adopted the convention that questions are summed over, whereas questions are weighted by the corresponding conditional probability .
The actions of players in a game give rise to a “probabilistic assignment”, a collection of probability distributions such that, for any pair of questions , is a probability distribution on pairs of answers to those questions. We may also represent as the rectangular matrix whose -th entry is . The value achieved by in the game is defined as
where we introduced a trace on the set of all by defining
In cases of interest the family of distributions is not arbitrary, but has a bipartite structure which reflects the bipartite nature of the game. Classical deterministic222Randomized strategies are convex combinations of deterministic strategies, thus a randomized strategy can always be replaced by a deterministic one achieving at least as high a value. strategies correspond to the case when for functions and taking the value exactly once. The functions and may be represented as vectors
respectively. is then the rank-one matrix , and we may express the value as
where the inner product is defined on by
We may similarly define an inner product on , and we will omit the subscripts when they are clear from context. Given a game matrix , we define its adjoint as the unique matrix such that for all and . Formally, if then .
Next we consider quantum strategies. A quantum strategy is specified by measurements for every and for every , where in general a measurement is any collection of positive semidefinite operators, of arbitrary finite dimension , that sum to identity. For any state representing the entanglement between the players,333In the literature the state is usually considered to be an integral part of the strategy. However it will be more convenient for us to not fix it a priori. Given measurement operators for both players in a game, it is always clear what is the optimal choice of entangled state; it is obtained as the largest eigenvector of a given operator depending on the game and the measurements (see below). this strategy gives rise to the family of distributions
This formula, dictated by the laws of quantum mechanics, corresponds to the probability that the players obtain outcomes , when performing the measurements , on their respective share of . One can check that positive semidefiniteness of the measurement operators together with the “sum to identity” condition imply that is a well-defined probability distribution on . To a quantum strategy we associate vectors
(Note that these definitions reduce to classical strategies whenever .) To express the success probability of this strategy in a game we extend the definition of the inner product as follows.
Definition 4 (Extended Inner Product).
The extended inner product
is defined, for and ,555Note the definition depends on a fixed choice of basis for the spaces and . by
With this definition the success probability of the strategy in can be expressed as
We also define the entangled value of the game, , to be the highest value achievable by any quantum strategy:
where here we slightly abuse notation and denote
We note that in the above the supremum may in general not be attained as optimal strategies may require infinite dimensions. In this paper we always restrict ourselves to finite dimensional strategies.666Thus when we say that achieve the value of we really mean that are finite-dimensional strategies whose value in can be made arbitrarily close to the optimum; for clarity we ignore this simple technicality in the whole paper.
It is well-known that any two-player game can be made into a projection game while essentially preserving its classical value. The following claim gives a partial extension of this fact to the case of the entangled value.
There exists a polynomial-time computable transformation mapping any two-player one-round game to a projection game such that the following hold:
In particular, if and only if , and . Moreover, for the entangled value we have the weaker bound
which implies .
Let be a game with (without loss of generality disjoint) question sets , , answer sets , , distribution on questions and acceptance predicate . Let be the projection game corresponding to the following scenario. The referee selects a pair of questions at random from , which it sends to the second player, and then sends either or to the first player, each with probability . Formally, is defined by question sets , , answer sets , , and a distribution given by , , and otherwise. For any and let be such that and if , and there is no valid answer for the first player if the second player’s answers are such that .
Then clearly is a projection game. Let be classical deterministic strategies for the players such that . Consider the strategy for in which answers as to questions and as to questions , and answers as . Then whenever the strategy provides answers to a pair of questions that satisfy the predicate the strategy gives answers to both and that are accepted in , hence
Conversely, let be a strategy for such that . Decompose into a pair of strategies in , depending on whether the question is or . The pair will give a rejected answer to a pair of questions only if gave a rejected answer to at least one of the questions and in . In the worst case the probability that provides rejected answers in is, say, fully concentrated on questions of the form . Hence
Finally, let be a pair of quantum strategies such that . To we unambiguously associate measurement operators for every , and for . Hence